The Role of Isometries in Cosmological Models

[摘要] This work is concerned with the role of symmetries in cosmological models in the context of the general theory of relativity. We review the observational evidence for the homogeneity and isotropy of our universe and discuss briefly the Friedmann-Robertson-Walker (FRW) models which describe homogeneous and isotropic cosmological models. The rest of the thesis provides an overview of those concepts of group theory and differential geometry relevant to the study of symmetry properties of homogeneous spacetimes. The techniques of differential geometry provide a method for describing the structures which can exist on a manifold. Once the manifold structure has been established it is possible to explore the symmetry properties of this manifold. The aspects of group theory relevant to the symmetry properties of a spacetime are then expounded. In particular we study connected Lie groups and their corresponding Lie algebras. The symmetry transformations that leave the metric invariant are called isometrics and the set of isometries form a group which can be split into a continuous component and a discrete component. The continuous isometries have associated with them infinitesimal isometries and these can be described by Killing vectors. These Killing vectors form the Lie algebra of the underlying symmetry group. The Killing vector fields therefore characterise the symmetry properties of the spacetime. The properties of isometries are discussed and some examples are given. In particular, to each Killing vector there corresponds a conserved quantity. The consequences of Lie group structure and the classification scheme for spatially homogeneous cosmologies (Bianchi classification) are outlined. We compute the Killing vector fields for the FRW models, discuss their algebraic properties and the conservation laws derivable from them. These can be used to derive simply and directly some of the familiar results of the Friedmann cosmologies.

[发布机构] University:University of Glasgow

[关键词] Theoretical physics